I can understand part of it. For a circular flow [itex]u_r=0[/tex] So the incompressibility yields that [itex] u_{\theta}[/itex] is not a function of theta. So the nonlinear term in the material derivative vanishes. But where does the last term [itex]-\frac{u_{theta}}{r^2} [/itex]
come from? And the entire first equation?!

1. By continuity, you must show that a 2-D circular motion for an incompressible fluid is only possible if the velocity field is independent of the angular coordinate;
[tex]\vec{v}=u_{\theta}(r,t})\vec{i}_{\theta}[/tex]
2.Write N-S as follows:
[tex]\frac{\partial\vec{v}}{\partial{t}}+(\vec{v}\cdot\nabla)\vec{v}=-\frac{1}{\rho}\nabla{p}+\nu\nabla^{2}\vec{v},\nabla=\vec{i}_{r}\frac{\partial}{\partial{r}}+\vec{i}_{\theta}\frac{\partial}{r\partial\theta},\nabla^{2}=\frac{\partial^{2}}{\partial{r}^{2}}+\frac{1}{r}\frac{\partial}{\partial{r}}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}[/tex]

Calculate the various terms this implies!
3) Show that the pressure must be independent of angle, since the pressure must be a continuous function.

Note:
"So the nonlinear term in the material derivative vanishes."
This is TOTALLY WRONG!!
You have curved motion.

Note:
"So the nonlinear term in the material derivative vanishes."
This is TOTALLY WRONG!!
You have curved motion.

But [itex]u_r[/itex] is zero for a circular flow so [itex]\vec{u}=(u_r,u_\theta)=(0,u_\theta)[/itex]. The nonlinear term in the material derivative is [itex](\vec{u} \cdot \nabla )\vec{u}=u_\theta \frac{\partial u_\theta}{r\partial \theta}[/itex]. And the continuity equation says: [itex]\nabla \cdot \vec{u}= \frac{\partial u_\theta}{r \partial \theta}=0[/itex]. So the nonlinear term vanishes, right?

Aha that explains a lot. Those darm changing unit vectors... But anyway, thanks a lot. The origin of the first equation is clear now, it is the r-component of N-S. As for the second equation; using the expression for the laplacian:

[tex]\frac{\partial^2 u_{\theta}}{\partial r^2} + \frac{1}{r} \frac{\partial u_{\theta}}{\partial r}+\frac{\partial ^2 u_{\theta}}{r^2 \partial \theta ^2}[/tex] Where the last term vanishes because [itex]u_{\theta}[/itex] is not a function of theta. So is the last term I mentioned in my first post a wrong?

Ouch, thats painful I hope I learned my lesson now... And about the independence of angle of the pressure. Isn't this obvious by symmetry? As the velocity profile does not depend on angle, there is no way to distinguish one direction form the other. The physical conditions are independent of the angle so the pressure must be as well?

It is very easy to prove that independence:
Look at the component equation in the angular direction.
Transfer the viscous term onto the other side; we gain therefore:
[tex]\frac{\partial{p}}{\partial\theta}=G(r,t)[/tex]
where G is some function solely of r and t, since the velocity is so (proven through continuity equation).

But this means, that the pressure must be, through integration:
[tex]p(r,\theta,t)=G(r,t)\theta+K(r,t)[/tex]

But, the polar coordinate points [tex](r,0)[/tex] and [tex](r,2\pi)[/tex] is the SAME point; hence, the pressure function must prescribe the same pressure value there.
(Alternatively, if you regard the angle to lie in the half-open interval [tex][0,2\pi)[/tex], the limiting value of the pressure when the angle tends towards [tex]2\pi[/tex] must be p(r,0,t); i.e, continuity of pressure)

We therefore must have:
[tex]p(r,0,t)=p(r,2\pi,t)\to{K}(r,t)=G(r,t)2\pi+K(r,t)\to{G}\equiv0[/tex]
Hence, p=K(r,t)=p(r,t), that is, independent of the angle.
QED.